Dipole-Dipole interaction
Of course the core of this library are the equations describing the dipole-dipole interaction and the collective decay
\[\begin{aligned} \Gamma_{ij} &= \frac{3}{2} \sqrt{\Gamma_i \Gamma_j} F_{ij}(k_a r_{ij}) \\ \Omega_{ij} &= \frac{3}{4} \sqrt{\Gamma_i \Gamma_j} G_{ij}(k_a r_{ij}) \end{aligned}\]
with
\[\begin{aligned} F_{ij}(\xi) &= \big( 1 - (\vec{e}^{(r)} . \vec{e}^{(d_{eg})})^2 \big) \frac{\sin \xi}{\xi} + \big( 1 - 3 (\vec{e}^{(r)} . \vec{e}^{(d_{eg})})^2 \big) \big( \frac{\cos \xi}{\xi^2} - \frac{\sin \xi}{\xi^3}\big), \\ G_{ij}(\xi) &= - \big(1 - (\vec{e}^{(r)} . \vec{e}^{(d_{eg})})^2 \big) \frac{\cos \xi}{\xi} + \big( 1 - 3 (\vec{e}^{(r)} . \vec{e}^{(d_{eg})})^2 \big) \big( \frac{\sin \xi}{\xi^2} - \frac{\cos \xi}{\xi^3}\big). \end{aligned}\]
They are implemented in the functions:
To create the interaction matrices the following two shortcuts are provided: